During the course of Futurama, most of our own solar system has been seen, and it looks like it’s changed considerably by the time the show takes place. Venus, the Moon, Mars, Neptune, Triton, and Pluto are all known to support animal life. The Sun is even inhabited in the video game, but I don’t know if that’s canonical.
Some inhabitants of the Sun did appear in “The Inhuman Torch,” but they were made of fire. Here’s a 2004 thread from a Futurama message board that mentions the references the show had given up to that point, and the Infosphere also covers the topic. I get the impression that the writers largely played with the popular conception of these worlds, which is why, for instance, there’s a penguin preserve on Pluto.
What’s the main thing people know about this recently demoted world? Well, obviously that it’s cold. Of course, it’s really too cold to support any kind of life, but the fact that we see a day-night cycle there in “The Birdbot of Ice-catraz” suggests that it’s been moved closer to the Sun. If Professor Farsworth can build a gravity pump that moves stars and Earth can be shifted in place by its robots all emitting exhaust at once, surely moving a dwarf planet wouldn’t be all that difficult. Leela does mention the temperature dropping to twenty degrees below absolute zero, a scientific impossibility (the word “absolute” is there for a reason), but that’s exactly what makes it funny. Incidentally, the fact that global warming has greatly affected Earth might be why people needed to move the penguins from Antarctica in the first place. We never learn whether McPluto is the same as this Pluto, but maybe McDonald’s maintains the penguin preserves as some kind of tax-deductible charity. Or they make the penguins into burgers. The Moon is also still much like it is today, even if it has an amusement park and a rural society of hydroponic farms. The surface is still covered with craters and gravity is still low, even though it should really be lower on Pluto and such is not demonstrated when the characters visit there.Oh, and it also has alligators in space helmets.
Al Gore has been introduced as Emperor of the Moon, but we don’t know exactly what this title entails. Saturn’s moon Titan being used for titanium mining is a play on words, since it’s not like there’s been any titanium discovered there (it’s mostly ice). In “The Sting,” Leela dreams of a garden on Venus, which could be a reference to how the planet was often depicted in science fiction before probes had explored it.
The fact that it was covered in clouds led many to speculate that it would be a lush, tropical world, and I’m sure its being named after the goddess of beauty only furthered that notion. We never find out if such gardens actually exist in the Futurama universe, but there are several references to Venus being inhabited. It’s been noted that the painting called the Venus de Venus shows a Blob much like the Horrible Gelatinous one, so that could be an indication that this is the dominant Venusian species, but again we don’t know for sure.

Mars is the non-Earth planet that’s been developed the most throughout the series.
It’s been established that it was terraformed into a place habitable by humans in the twenty-seventh century, and even has at least one jungle. We don’t know exactly when Sir Reginald Wong bought the planet from the Native Martians in exchange for a bead that turned out to be an enormous diamond, but since the natives were presumably living there before Terran species were brought there, they probably don’t breathe oxygen. Most of them leave the planet in “Where the Buggalo Roam” upon learning that they’re rich, but some remain behind to work in Mars Vegas.
In “Viva Mars Vegas,” Amy Wong gives the tribe the rights to her family’s casino. While it initially appears that the events of “A Farewell to Arms” have led to the destruction of the planet (an event the Native Martians had predicted well ahead of time), it actually survives intact in a different orbit. Mars is also the native world of the buggalo, giant insects with many bovine traits, including the ability to produce milk. Well, a milk-like substance, anyway, but you have to take what you can get when cows have gone extinct.

Oddly, the other planet we’ve seen a significant amount is Neptune, which in real life is a gas giant. If it was terraformed, it must have been a much more difficult task than it was with Mars. It’s primarily inhabited by four-armed humanoids who eat slugs, and is the base of operations for the robotic Santa Claus.
It’s also home to yetis, a species also known to live on the moon Triton.Totally different from sasquatches, apparently.
It’s possible that these life forms arrived on Neptune from somewhere else, but we’re never really told. When we see the characters visiting this planet, not only does it appear solid, but its gravity never looks to be an impediment to normal movement.

Other known places within our galaxy are also seen or mentioned from time to time. The plot of “When Aliens Attack” required a location about 1000 light-years from Earth, so the writers went with Omicron Persei.
The fact that the inhabitants call their own world Omicron Persei 8 is an example of the geocentric trend in sci-fi, but then they also speak English amongst themselves. The third world from the star Antares is where electronic waste is taken for recycling. The Cygnoids presumably come from a planet orbiting a star in the constellation Cygnus, but we don’t know which one.